Found problems: 2023
2021 JBMO TST - Turkey, 8
$w_1$ and $w_2$ circles have different diameters and externally tangent to each other at $X$. Points $A$ and $B$ are on $w_1$, points $C$ and $D$ are on $w_2$ such that $AC$ and $BD$ are common tangent lines of these two circles. $CX$ intersects $AB$ at $E$ and $w_1$ at $F$ second time. $(EFB)$ intersects $AF$ at $G$ second time. If $AX \cap CD =H$, show that points $E, G, H$ are collinear.
1952 Miklós Schweitzer, 2
Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?
2006 Australia National Olympiad, 1
In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.
2010 District Olympiad, 1
A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c$ and $ \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q$.
i) Prove that:
\[ a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC}\]
ii) Show that $ a\equal{}bp\plus{}cq$.
iii) If $ a^2\equal{}4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrents.
2011 IMAC Arhimede, 2
Let $ABCD$ be a cyclic quadrilatetral inscribed in a circle $k$. Let $M$ and $N$ be the midpoints of the arcs $AB$ and $CD$ which do not contain $C$ and $A$ respectively. If $MN$ meets side $AB$ at $P$, then show that
$\frac{AP}{BP}=\frac{AC+AD}{BC+BD}$
1979 IMO Longlists, 74
Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.
2008 Romania Team Selection Test, 3
Let $ ABCDEF$ be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles $ ACE$ or $ BDF$ is at least 1.
2010 Baltic Way, 12
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.
$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.
$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.
2005 District Olympiad, 3
Let $ABC$ be a non-right-angled triangle and let $H$ be its orthocenter. Let $M_1,M_2,M_3$ be the midpoints of the sides $BC$, $CA$, $AB$ respectively. Let $A_1$, $B_1$, $C_1$ be the symmetrical points of $H$ with respect to $M_1$, $M_2$ and $M_3$ respectively, and let $A_2$, $B_2$, $C_2$ be the orthocenters of the triangles $BA_1C$, $CB_1A$ and $AC_1B$ respectively. Prove that:
a) triangles $ABC$ and $A_2B_2C_2$ have the same centroid;
b) the centroids of the triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ form a triangle similar with $ABC$.
2012 European Mathematical Cup, 1
Let $ABC$ be a triangle and $Q$ a point on the internal angle bisector of $\angle BAC $. Circle $\omega_1$ is circumscribed to triangle $BAQ$ and intersects the segment $AC$ in point $P \neq C$. Circle $\omega_2$ is circumscribed to the triangle $CQP$. Radius of the cirlce $\omega_1$ is larger than the radius of $\omega_2$. Circle centered at $Q$ with radius $QA$ intersects the circle $\omega_1$ in points $A$ and $A_1$. Circle centered at $Q$ with radius $QC$ intersects $\omega_1$ in points $C_1$ and $C_2$. Prove $\angle A_1BC_1 = \angle C_2PA $.
[i]Proposed by Matija Bucić.[/i]
2000 Bundeswettbewerb Mathematik, 3
A convex quadrilateral $ABCD$ is inscribed in a semicircle with diameter $AB$. The diagonals $AC,BD$ intersect at $S$, and $T$ is the projection of $S$ on $AB$. Show that $ST$ bisects angle $CTD$.
2010 International Zhautykov Olympiad, 3
Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that
1) $\angle OIH > 90^0$(2 points)
2)$\angle OIH >135^0$(7 points)
balls for 1) and 2) not additive.
2006 Austrian-Polish Competition, 6
Let $D$ be an interior point of the triangle $ABC$.
$CD$ and $AB$ intersect at $D_{c}$,
$BD$ and $AC$ intersect at $D_{b}$,
$AD$ and $BC$ intersect at $D_{a}$.
Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that
$(AD_{c}D) = (KH_{m}H)$
$(BD_{c}D) = (LH_{m}H)$
$(BD_{a}D) = (LH_{k}H)$
$(CD_{a}D) = (MH_{k}H)$
$(CD_{b}D) = (MH_{l}H)$
$(AD_{b}D) = (KH_{l}H)$
where $(PQR)$ denotes the area of the triangle $PQR$
2012 Romanian Master of Mathematics, 2
Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$.
[i](United Kingdom) David Monk[/i]
2010 Indonesia TST, 4
Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r\plus{}r_0\equal{}R.\]
[i]Soewono, Bandung[/i]
2010 Baltic Way, 13
In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$, determine all possible values of $\angle CAB$.
2015 Kazakhstan National Olympiad, 6
The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?
2006 MOP Homework, 3
In triangle $ ABC$,$ \angle BAC \equal{} 120^o$. Let the angle bisectors of angles
$ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively.
Prove that the circle on diameter $ EF$ passes through $ D.$
2016 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$.
Prove that $\angle QCB = \angle PCO$.
2008 Hong Kong TST, 3
Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE \cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC \cap BD \equal{} E'$. Suppose also that $ (ABD')\cap (AC'E) \equal{} A''$, $ (BCE')\cap (BD'A) \equal{} B''$, $ (CDA')\cap (CE'B) \equal{} C''$, $ (DEB')\cap DA'C \equal{} D''$ and $ (EAC')\cap (EB'D) \equal{} E''$. Prove that $ AA''$, $ BB''$, $ CC''$, $ DD''$ and $ EE''$ are concurrent.
2009 Indonesia TST, 3
Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$.
(a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point.
(b) Find the the locus of point $ E$.
2009 Romania Team Selection Test, 2
Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.
2012 Macedonia National Olympiad, 4
A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.
2014 ELMO Shortlist, 13
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
2002 Romania National Olympiad, 1
In the Cartesian plane consider the hyperbola
\[\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} \]
and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$
For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.